Isotachic train

The graphical interpretation of Eq. (16-199) is shown in Fig. 16-37 for the conditions of Example 16-14. An operating line is drawn from the origin to the point of the pure displacer isotherm at c1 = cf. For displacement to occur, the operating line must cross the pure component isotherms of the feed solutes. The product concentrations in the isotachic train are found where the operating line crosses the isotherms. 

Following elution of the isotachic train and the displacer solution from the column, the column must be regenerated and reequilibrated with the carrier before any subsequent displacement separation. This reequilibration step can be lengthy and is frequently considered a major Umitation to efficient displacement operation. Displacement chromatography requires the competitive isotherms of the solutes and the displacer to be convex upward and to not intersect each other. (See the entry Distribution Coefficient for related information.)... 

This effect is fundamental in displacement chromatography (see Chapter 9) and explains the displacement of the band of one component by another band and the eventual formation of an isotachic train. 

Steady State in the Displacement Mode. The Isotachic Train...439... 

Critical Column Length for Isotachic Train Formation.461... 

Figure 9.3 Illustration of the operating line and determination of the plateau concentration of the individual bands, (a) Single-component isotherms with Q > Q cnt (Eq. 9.7). (b) Profiles of the zones in an ideal isotachic train.

It is not possible to adjust the band heights for each component independently. Eor a given concentration of the displacer, the band height or plateau concentration of each sample component at steady state is constant. Therefore, in order to conserve the area of each band, which must remain equal to the area of the injected profile, the width of each band is proportional to the amoimt of the corresponding compoxmd injected. As the amotmt of any feed component injected in the column increases, the width of its band in the isotachic train increases (Figures 9.4a and 9.4b). If the amounts of several components in the sample increase, the widths of their bands increase at constant height (Figures 9.4b and 9.4c). 

Although the previous discussion supplies a correct description of the isotachic train in the ideal model and of its steady propagation, it does not provide any clue regarding the individual band profiles during the progressive formation... 

The watershed point of component i is the displacer concentration (C = w ) corresponding to the intersection of the displacer isotherm and the initial tangent to the isotherm of that component. This concept was first identified by Glueckauf [6,7]. This critical concentration is given by Eq. 9.7. No displacement of component i is possible if the displacer concentration is below its watershed point (Figure 9.5a). Under conditions where C = w, the tail of the first component will end at the front of the second component band. As the displacer concentration falls below the watershed point, the rear boimdary of the first component separates completely from the isotachic train. In this case, which is not exceptional, one or a few early eluting components appear as independent elution bands before the isotachic train. This may be impossible to avoid—for example, if the solubility of the displacer in the carrier is insufficient. 

Note that the relationship between the isotherms of the feed components and the operating line implies that the successive bands of the isotachic train should have increasing heights (in terms of concentration because the detector response may vary widely from component to component, the actual band heights may... 

Antia and Horvath [32] have provided an analysis of the isotachic patterns in displacement chromatography that take into accormt the effects of a selectivity reversal. They used for this purpose the multicomponent isotherm generated by the IAS theory. In order to examine the possibility of performing separations even when the single-component isotherms of two components intersect and cross over, they analyzed the stability of the displacement train and the criteria for displacement and for the stability of the resulting botmdaries that are established in an isotachic train. 

Since the transformation of C, into co defined by Eq. 9.15 is independent of both X = z/L and t = ut/L, the solution scheme is applicable to any point in the (x, t) plane if an initial discontinuity is imposed at that point. If discontinuities are imposed at two or more points along the x or the t axis, the solution may be derived by constructing separately the wave solutions for each of these points. However, after a finite period of time, any two wave solutions centered in two different, but adjacent, points will meet each other. An overlap region appears, where the solution is influenced by both sets of data. Such a phenomenon is called wave interaction. From what was explained in the previous section, this interaction takes place during the formation of the isotachic train of any mixture. 

Figure 9.8 The distance-time diagram illustrating the progressive formation of an isotachic train for a binary mixture. H.-K. Rhee and N. R. Amundson, AICHE J, 28 (1982) 423 (Fig. 2). Reproduced by permission of the American Institute of Chemical Engineers. 19S2 AIChE. All rights reserved.

As the displacer concentration increases, co decreases (Eq. 9.27), and it moves to the left-hand side in Eq. 9.28. The number of shock waves increases. If we are interested in obtaining an isotachic train after the development is over, we want to obtain n rear shocks. This is possible only if the displacer concentration is large enough, and co becomes the first in the ordered set of characteristic parameters in Eq. 9.28. This requires that co < fli. Introducing this condition in Eq. 9.27 gives... 

As we have already shown, using a simple approach based on the assumption that a steady state can be reached (Eqs. 9.6 and 9.7), the critical displacer concentration for successful displacement of a multicomponent mixture depends only on the adsorption isotherm of the displacer On and bn) and on the initial slope of the isotherm of the less retained solute at infinite dilution (fli). Qn the other hand, if the displacer concentration is so small that co is the last element in the ordered set of characteristic parameters of the displacement (Eq. 9.28), there are no shocks, and no isotachic train is possible. This happens when co > a i. Combining this condition and Eq. 9.27 gives... 

In this case, we observe the successful formation of an isotachic train including n — y bands, preceded by a series of — 1 bands eluted separately or interfering, but not included in an isotachic train. The /th wave is a shock if 07 < cof and a simple wave if a "... 

In the final stage of the development of an isotachic train, we obtain n plateaus and as many shock waves one plateau and one shock wave for each pure band y. Since only component / is present in the /th band, and all the other components are missing from that band ... 

Since the paths of the n shocks are now known, we can locate each of them at the final stage of the isotachic train without the need for a detailed analysis of the interactions between the bands. The shocks are located using Eqs. 9.41 to 9.43. 

In order to calculate the minimum bed length required for the formation of an isotachic train, however, we must analyze all the interactions involved. In addition, if the concentrations of the solutes are chosen so that to is larger than for one or more components, the corresponding waves of these solutes are simple waves, and we must also analyze the successive interactions between these simple waves and each of the shocks in order to obtain the solution [10]. An excellent... 

We show in Figures 9.10 the separation of a ternary mixture (solutes Ai, A2 and A3), displaced by a solution of A4, as predicted by the ideal model and calculated by Rhee et al. [10]. In this case, the displacer concentration is higher than the critical value, to < a, and the isotachic train finally formed includes the bands of all three components. For clarity, the authors have shown two separate chromatograms for each time at which they calculated the individual band pro-... 

Even in an ideal coltunn, the reorganization of the distribution of the component concentrations between the injection and the formation of the isotachic train requires a certain time, i.e., it cannot be achieved in less than a minimum migration distance. This distance can be derived from the distance-time diagram. 

From a theoretical viewpoint, frontal analysis and displacement chromatography are important and interesting problems because there are as5onptotic solutions for the breakthrough curves of frontal analysis and for the band profiles in the isotachic train in displacement chromatography. An asymptotic solution is an analytical solution obtained after an infinite migration distance. The existence... 

We discuss in this first part the formation of the isotachic train using the equilibrium-dispersive model and the influence of the various parameters that control the characteristics of this train the displacer concentration, the sample size, the column length, the concentration of the feed, and the column efficiency. The results differ from those reported in Chapter 9, which were obtained with the ideal model in which there is no dispersion. Because the differences observed consist essentially in the formation of mixed zones between the bands in the isotachic train, many results remain similar. We also discuss the behavior of trace components, either those contained in the sample or those contained in the displacer. 

We compare in Figure 12.1a the isotachic trains calculated with the ideal and the equilibrium-dispersive models under the experimental conditions given in Chapter 9. The major difference observed concerns the boundary between adjacent zones. These boundaries are no longer vertical fronts or shocks they have become shock layers (Chapter 16, Section 16.1.4). The shock layer thickness is quite significant compared to the natural width of these zones, and it increases... 

Figure 12.1 Comparison of the concentration profiles of the bands in an isotachic train predicted by the ideal model and the equilibrium-dispersive model, (a) Displacement chromatogram. Profiles predicted by the ideal model (black lines) and by the equilibrium-dispersive model (thick shaded line). Parameters e = 0.80 fp = 1 mL/min mi = 50 mg m2 = 75 mg m3 = 100 mg Crf = 100 mg/mL L = 50 cm ai = 16 2 = 20 fls = 24 = 30 bi =...

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